Embedded newform 403.2.be.c.57.1

• Label: 403.2.be.c.57.1
• Level: $403$
• Weight: $2$
• Character: 403.57
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.be (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(34\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			57.1
Character	\(\chi\)	\(=\)	403.57
Dual form			403.2.be.c.99.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.92370 + 1.92370i) q^{2} +(2.94175 + 1.69842i) q^{3} -5.40127i q^{4} +(-0.883813 - 0.236817i) q^{5} +(-8.92630 + 2.39180i) q^{6} +(-2.87039 + 0.769118i) q^{7} +(6.54303 + 6.54303i) q^{8} +(4.26925 + 7.39456i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 4 q^{2} - 24 q^{3} + 2 q^{5} - 36 q^{6} - 2 q^{7} - 8 q^{8} + 60 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.92370	+	1.92370i	−1.36026	+	1.36026i	−0.486688	+	0.873576i	\(0.661795\pi\)
							−0.873576	+	0.486688i	\(0.838205\pi\)
\(3\)	2.94175	+	1.69842i	1.69842	+	0.980582i	0.947263	+	0.320456i	\(0.103836\pi\)
							0.751155	+	0.660126i	\(0.229497\pi\)
\(5\)	−0.883813	−	0.236817i	−0.395253	−	0.105908i	0.0557174	−	0.998447i	\(-0.482255\pi\)
							−0.450971	+	0.892539i	\(0.648922\pi\)
\(7\)	−2.87039	+	0.769118i	−1.08490	+	0.290699i	−0.756603	−	0.653874i	\(-0.773143\pi\)
							−0.328301	+	0.944573i	\(0.606476\pi\)
\(11\)	0.594095	+	0.159187i	0.179126	+	0.0479967i	0.347267	−	0.937766i	\(-0.387110\pi\)
							−0.168141	+	0.985763i	\(0.553776\pi\)
\(13\)	−1.92836	+	3.04654i	−0.534832	+	0.844958i
							\(17\)	−0.496085	+	0.859244i	−0.120318	+	0.208397i	−0.919893	−	0.392169i	\(-0.871725\pi\)
0.799575	+	0.600566i	\(0.205058\pi\)
\(19\)	0.631367	+	2.35629i	0.144846	+	0.540571i	0.999762	+	0.0218057i	\(0.00694151\pi\)
							−0.854917	+	0.518765i	\(0.826392\pi\)
\(23\)	3.94176			0.821914			0.410957	−	0.911655i	\(-0.365195\pi\)
							0.410957	+	0.911655i	\(0.365195\pi\)
\(29\)		−	0.574476i		−	0.106677i	−0.998576	−	0.0533387i	\(-0.983014\pi\)
							0.998576	−	0.0533387i	\(-0.0169863\pi\)
\(31\)	−3.51216	−	4.32027i	−0.630802	−	0.775944i
							\(37\)	0.302302	+	1.12821i	0.0496981	+	0.185476i	0.986313	−	0.164886i	\(-0.0527255\pi\)
−0.936615	+	0.350361i	\(0.886059\pi\)
\(41\)	2.32358	+	0.622601i	0.362882	+	0.0972340i	0.435653	−	0.900115i	\(-0.356518\pi\)
							−0.0727708	+	0.997349i	\(0.523184\pi\)
\(43\)	−3.98567	+	6.90338i	−0.607809	+	1.05276i	0.383792	+	0.923419i	\(0.374618\pi\)
							−0.991601	+	0.129336i	\(0.958715\pi\)
\(47\)	0.387432	+	0.387432i	0.0565128	+	0.0565128i	0.734798	−	0.678286i	\(-0.237277\pi\)
							−0.678286	+	0.734798i	\(0.737277\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.be.c.57.1	✓	136
13.8	odd	4	inner	403.2.be.c.398.1	yes	136
31.6	odd	6	inner	403.2.be.c.161.1	yes	136
403.99	even	12	inner	403.2.be.c.99.1	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.be.c.57.1	✓	136	1.1	even	1	trivial
403.2.be.c.99.1	yes	136	403.99	even	12	inner
403.2.be.c.161.1	yes	136	31.6	odd	6	inner
403.2.be.c.398.1	yes	136	13.8	odd	4	inner